Transformation monoids can be naturally used to recognize trace languages. We say that a
trace language \(L \subseteq TR(\widetilde{\Sigma })\)  is recognized by a tm \(T=(X,M)\)  if there
exists a morphism \(\varphi \colon TR(\widetilde{\Sigma })\rightarrow T\) , an initial element
\(x_{\textrm {in}} \in X\)  and a final subset \(X_{\textrm {fin}} \subseteq X\)  such that
\(L = \lbrace t \in TR(\widetilde{\Sigma })\mid \varphi (t)(x_{\textrm {in}}) \in X_{\textrm {fin}}\rbrace \) . A trace
language is said to be recognizable if it is recognized by a finite tm (see
[1]}, [2]}).
